Mustafa Aksu, with AI collaborators (Claude, ChatGPT, Gemini, Grok)
January 2026
Contents
1. Abstract
We demonstrate computationally the emergence of spacetime-like geometry from a pre-geometric network of coupled
oscillators in the Relational Time Geometry (RTG) framework. Starting from a minimal 4-node dipole (vacuum seed),
a flux-balanced growth algorithm minimizes boundary charge while using a controlled “breathing” mechanism
(temporary charge excursions) to avoid local growth stalls. In a representative near-vacuum growth run we achieve
\(97.7\%\) boundary charge neutralization, with \(Q\) improving from \(Q=-1.60\) to \(Q=-0.04\) at \(N=16\) triads
(\(14\) nodes).
A Hodge decomposition of the edge residual field \(z\) reveals that \(75.2\%\) of curvature variance is
cohomologically irreducible, establishing a topological obstruction to exact flatness (“geometric floor”).
The resulting structure is neither flat nor empty: it is a quantum foam—a discrete relational analog
of Wheeler’s spacetime foam—characterized by boundary neutrality together with persistent internal curvature
fluctuations.
The results support RTG’s central hypothesis: geometry arises from frustrated frequency synchronization.
They also suggest a Machian feature of the model: coherent geometry nucleates around relational defects (seeded
structures) rather than from an empty background.
(Note: this “void fails” observation was seen informally during development; systematic tests of random seeding
remain future work.)
2. Theoretical Framework
2.1 RTG Postulates
- No background geometry exists a priori.
- Each node \(i\) carries intrinsic dynamical variables: frequency \(\omega_i\) and phase \(\phi_i\).
- Emergent distance law: relational distance is derived from frequency incompatibility,
commonly expressed as
\[
r_{ij} = \frac{\kappa}{|\Delta\omega_{ij}|}, \quad \Delta\omega_{ij}=\omega_i-\omega_j,
\]
where \(\kappa\) sets units. - Curvature lives on edges through a logarithmic residual field \(z_{ij}\):
\[
z_{ij}=\ln\!\big(r^{\mathrm{em}}_{ij}\big)-\ln\!\big(s\,r^{\mathrm{geo}}_{ij}\big),
\]
where \(r^{\mathrm{em}}_{ij}\) is frequency-derived distance, \(r^{\mathrm{geo}}_{ij}\) is Euclidean embedding
distance, and \(s\) is a global scale fitted in the bootstrap stage.
2.2 Two Independent Boundary Metrics
Throughout the ESM study we distinguish two boundary observables, computed only on boundary edges \(e\in\partial\mathcal{K}\)
of the current cluster \(\mathcal{K}\):
- Boundary charge (signed):
\[
Q(\mathcal{K})=\sum_{e\in\partial\mathcal{K}} z_e.
\]
- Boundary roughness (magnitude):
\[
C(\mathcal{K})=\sqrt{\left\langle z_e^2 \right\rangle_{e\in\partial\mathcal{K}}}
=\sqrt{\frac{1}{|\partial\mathcal{K}|}\sum_{e\in\partial\mathcal{K}} z_e^2 }.
\]
These metrics are independent: a cluster can be neutral-but-rough (\(Q\approx 0\), \(C\) large) or smooth-but-charged
(\(C\) small, \(Q\neq 0\)). This duality is central to the quantum-foam interpretation.
3. Methodology
Phase I: The Geometric Floor (Hodge Limit)
We generate random Delaunay-like graphs in 3D and fit node frequencies \(\omega_i\) to minimize the edge residuals \(z_{ij}\)
using robust optimization (IRLS-style procedures). Optimization consistently stalls at a non-zero residual “floor,” motivating
a topological decomposition of the \(z\)-field rather than treating the floor as numerical error.
Representative floor values in the embedding experiments:
- Median \(|z|\approx 0.46
- Mean \(|z|\approx 0.21
We decompose the edge field using Hodge decomposition:
\[
z = d f + \delta g + h,
\]
where \(d f\) is the exact (gradient) component, \(\delta g\) is the coexact (rotational) component, and \(h\) is harmonic.
In a representative run, the variance split was:
- Exact: \(24.8\%\) (removable)
- Coexact + harmonic: \(75.2\%\) (irreducible)
Conclusion: the geometric floor is cohomologically protected; exact flatness is topologically obstructed
by discrete connectivity.
Phase II: The Vacuum Seed (Dipole)
The ESM cluster is built from triads (triangles / 2-simplices). A single triad typically behaves as a
“charged” element (unscreened boundary contribution). Growth from a charged seed tends to be unstable.
We therefore construct a dipole (vacuum seed): two triads sharing an edge with opposing boundary contributions,
producing a locally screening configuration.
Seed selection uses Pareto filtering and stability constraints:
- Holonomy / residual variance threshold (geometric coherence)
- Hessian stability: positive eigenvalues and adequate minimum stiffness
- Orientation compatibility: shared edge sign structure consistent
- Collective stability: paired configuration stability (combined local Hessian)
The dipole seed is a 4-node “butterfly” topology (schematic):
[a]
/ \ [b]—[c] \ /
[d]
The seed’s boundary charge \(Q\) is computed on its exposed boundary edges (not by summing per-triad fluxes).
Phase III: Flux-Balanced Growth (Stokes Theorem Analog)
The pivotal correction enabling robust growth is the Stokes-like realization that “total flux” must be computed on the
boundary only.
Old (incorrect, double-counts internal edges):
\[
\Phi_{\mathrm{old}}=\sum_{t\in\mathcal{K}} \phi(t)
\]
New (correct boundary charge):
\[
Q(\mathcal{K})=\sum_{e\in\partial\mathcal{K}} z_e
\]
When a triad attaches across a boundary edge \(b\), that edge becomes internal and two new boundary edges appear. The boundary
charge updates locally as:
\[
\Delta Q = -z_b + z_{e_1} + z_{e_2}.
\]
Growth step selection:
1) Enumerate attachable candidates along boundary edges.
2) Filter candidates by stability (Hessian) and Pareto constraints (local z-variance).
3) Score candidates by expected boundary metrics (Q and C) and select the best.
4) Update the cluster boundary state.
Breathing mechanism (key novelty): the algorithm allows temporary increases in \(|Q|\) (“inhaling”) to cross
geometric bottlenecks, provided subsequent steps reduce \(|Q|\) (“exhaling”). This prevents stalling in local “vacuum prisons.”
Visual placeholder: Growth dynamics (boundary charge \(Q\) across steps, and joint evolution of \(Q\) and \(C\)).
Phase IV: Refinement (Swap + Surgery) and DRR
After growth, we refine the cluster using three mechanisms:
- Swap-explore: stochastic boundary swaps to reduce roughness while respecting a charge cap \(|Q|\le Q_{\max}\).
- k-exchange surgery: remove \(k\) triads and add \(k\) different triads to escape local minima.
- Dynamic Resonance Relaxation (DRR): local updates to \(\omega_i\) on cluster nodes, followed by recomputing \(z\), to allow the structure to self-settle.
A critical empirical observation is that aggressive smoothing (especially surgery with strong roughness penalties) can produce
fragmentation, quantified by increased connected components. We track this with topological diagnostics such as
\(\beta_0\) (number of connected components) and the number of boundary components.
Visual placeholder: Topology audit output for representative clusters (boundary components, \(\beta_0\), \(\beta_1\), and interior holonomy).
4. Results Summary
4.1 Headline Numbers
- Boundary screening: \(Q=-1.60 \rightarrow Q=-0.04\) at \(N=16\) triads (\(97.7\%\) neutralization).
- Geometric floor: Hodge decomposition shows \(75.2\%\) irreducible curvature variance.
4.2 Representative Cluster Diagnostics
The study produced multiple clusters across sizes. The following are representative checkpoints that illustrate
screening, roughness, and topology together.
| Checkpoint | Size | Boundary metrics | Topology diagnostics | Interpretation |
|---|---|---|---|---|
| Near-vacuum demonstration | \(N=16\) triads (\(\approx 14\) nodes) | \(Q=-0.04\), \(97.7\%\) neutralization; representative \(C\approx 0.68\) | (Topology depends on realization; used as primary screening demonstration) | Strong macroscopic screening with non-zero roughness |
| Shell checkpoint | \(N=35\) triads | \(Q=-0.0614\), \(C=0.1678\) | \(\beta_0=4\), boundary components \(=4\), \(\beta_1=0\) | Roughness reduced; cluster already multi-component |
| “Over-polish” example | \(N=35\) triads (refined) | \(Q=+0.0675\), \(C=0.1575\) | \(\beta_0=6\), boundary components \(=7\), \(\beta_1=0\) | Smoother boundary but increased fragmentation (debris) |
| Mesoscopic checkpoint | \(N=70\) triads | \(Q=-0.0008\), \(C=0.1702\) | \(\beta_0=5\), \(\beta_1=2\), boundary components \(=8\) | Near-neutral boundary with interior tunnels/holes (foam-like topology) |
These checkpoints illustrate the central trade-off discovered during refinement:
reducing \(C\) can increase \(\beta_0\), i.e., connectivity loss.
4.3 The Quantum Foam Signature
We use the term quantum foam in a precise operational sense:
- Boundary neutrality: \(Q(\mathcal{K}) \approx 0\) (screened boundary; “flat” to an external observer).
- Irreducible interior structure: non-zero roughness and topological/holonomy diagnostics consistent with an irreducible curvature floor.
5. Physical Interpretation: The Machian Vacuum
5.1 The “Stage Without Actors” Paradox
The ESM results suggest a Machian framing: geometry does not appear as an empty container. Instead, coherent geometry
nucleates around relational defects (structured seeds).
- The void fails (informal observation): without a structured seed, growth often stalls and does not reliably produce coherent geometry.
- The seed is the anchor: the dipole seed provides a relational reference that the vacuum screens, analogous to “fixed stars” anchoring inertial structure in Mach’s principle.
- Conclusion: in RTG/ESM, “space” and “matter/defect” are co-defining: spacetime is the relaxation field of relational defects.
5.2 Screening Analogy (QCD-Inspired)
A useful analogy is screening in gauge theories: local charges bind and screen to produce macroscopic neutrality.
| Concept | ESM / RTG | QCD analogy |
|---|---|---|
| Microscopic charge | Curvature residual on boundary edges | Color charge |
| Local screening unit | Dipole (vacuum seed) | Meson-like neutral pair |
| Macroscopic neutrality | \(Q \approx 0\) on the boundary | Color neutrality of hadrons |
| Persistent microstructure | Non-zero \(C\), irreducible curvature variance | Non-trivial vacuum structure |
5.3 Defects and Proto-Gravity (Deferred to Future Tests)
The model naturally defines “matter” as unscreened curvature:
vacuum corresponds to \(Q\approx 0\), while defects correspond to \(Q\neq 0\).
Whether interactions between multiple defects yield an effective force law is an open experimental target (see Future Directions).
6. Limitations and Scope
- Scale: mesoscopic clusters were grown up to \(N\approx 200\) triads in development runs, but systematic scaling laws and continuum limits are not yet established.
- Dynamics: the reported results focus on static cluster construction and refinement; time evolution of the full RTG dynamics is future work.
- Statistics: interior holonomy audits at small \(N\) can have low statistical power (few interior samples); larger clusters are required for robust bulk statistics.
- Non-uniqueness: many locally stable configurations exist; identifying a unique ground state or ensemble requires thermodynamic analysis.
- Dimensionality: embeddings were performed in 3D; extension to a fully dynamical \(3+1\) construction remains open.
7. Conclusions
- Stokes-like correction enabled growth: computing charge on the boundary,
\(Q=\sum_{e\in\partial\mathcal{K}} z_e\), was the turning point for robust cluster growth. - Screening works: flux-balanced growth achieved \(97.7\%\) boundary neutralization in a near-vacuum cluster (\(Q=-1.60 \rightarrow -0.04\)).
- Flatness is obstructed: Hodge decomposition shows \(75.2\%\) irreducible curvature variance, supporting the quantum-foam interpretation.
- Two metrics are required: neutrality (\(Q\)) and smoothness (\(C\)) are independent and must be optimized jointly.
- Refinement has a stability window: aggressive smoothing can reduce \(C\) but increase fragmentation (\(\beta_0\uparrow\)).
8. Future Directions
- Mesoscopic scaling and shell closure (“dipole magic numbers”):
test stability peaks at sizes predicted by symmetric dipole shells. One geometric hypothesis is that closure occurs at
\[
N_{\mathrm{dipole}} = 2\,P_n,\quad P_n=\binom{n+3}{4}\in\{1,5,15,35,70,126,210,\dots\},
\]
yielding \(N_{\mathrm{dipole}}\in\{2,10,30,70,140,252,420,\dots\}\). These targets are motivated by complete bilayers
around a dipole seed (symmetry completion), not by numerology. - Defect dynamics: inject a charged triad (controlled \(Q\neq 0\)) into an otherwise neutral foam and measure the spatial response of boundary charge and roughness.
- Proto-gravity test: place two defects at controlled separation and test whether interference of screening fields produces an effective attraction (including whether any regime approximates an inverse-square law).
- Thermodynamics: define an ensemble over clusters and measure entropy of curvature configurations to obtain an equation of state for the vacuum.
- Time evolution: extend from static growth/refine to dynamic RTG evolution with phases \(\phi_i(t)\) and frequencies \(\omega_i(t)\) co-evolving.
Appendix: Code Pipeline and Key Equations
Appendix A: Code Pipeline
esm_bootstrap.py: mesh generation and \(\omega\) fitting (including global scale \(s\))esm_dipole_search.py: vacuum seed candidate search (Pareto + stability filters)esm_dipole_lock.py: seed locking and validationesm_flux_grow_v11_full.py: flux-balanced growth + refinement + DRResm_topology_audit_v3.py: \(\beta_0,\beta_1\), boundary component diagnostics, and interior holonomy auditsesm_prune_largest_component.py: optional post-processing for fragmentation (keep the largest component)esm_visualize_cluster.py: cluster visualization utilities
Appendix B: Key Equations
Emergent distance law:
\[
r_{ij}=\frac{\kappa}{|\omega_i-\omega_j|}.
\]
Edge residual (“curvature”):
\[
z_{ij}=\ln\!\big(r^{\mathrm{em}}_{ij}\big)-\ln\!\big(s\,r^{\mathrm{geo}}_{ij}\big).
\]
Boundary charge (Stokes analog):
\[
Q(\mathcal{K})=\sum_{e\in\partial\mathcal{K}} z_e.
\]
Boundary roughness:
\[
C(\mathcal{K})=\sqrt{\frac{1}{|\partial\mathcal{K}|}\sum_{e\in\partial\mathcal{K}} z_e^2 }.
\]
Hodge decomposition:
\[
z = d f + \delta g + h.
\]
Typical parameter choices in the reported pipeline include \(\kappa=1.0\) (units) and a fitted global scale \(s\) on the order
of \(10^3\) (e.g., \(s\approx 6.5\times 10^3\) in a representative run).
Document Version 2.2 — Finalized January 2026